Equivariant Semidefinite Lifts of Regular Polygons

Hamza Fawzi (), James Saunderson () and Pablo A. Parrilo ()
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Hamza Fawzi: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom
James Saunderson: Department of Electrical and Computer Systems Engineering, Monash University, Victoria 3800, Australia
Pablo A. Parrilo: Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Mathematics of Operations Research, 2017, vol. 42, issue 2, 472-494

Abstract: Given a polytope P ⊂ ℝ n , we say that P has a positive semidefinite lift (psd lift) of size d if one can express P as the projection of an affine slice of the d × d positive semidefinite cone. Such a representation allows us to solve linear optimization problems over P using a semidefinite program of size d and can be useful in practice when d is much smaller than the number of facets of P . If a polytope P has symmetry, we can consider equivariant psd lifts, i.e., those psd lifts that respect the symmetries of P . One of the simplest families of polytopes with interesting symmetries is regular polygons in the plane. In this paper, we give tight lower and upper bounds on the size of equivariant psd lifts for regular polygons. We give an explicit construction of an equivariant psd lift of the regular 2 n -gon of size 2 n − 1, and we prove that our construction is essentially optimal by proving a lower bound on the size of any equivariant psd lift of the regular N -gon that is logarithmic in N . Our construction is exponentially smaller than the (equivariant) psd lift obtained from the Lasserre/sum-of-squares hierarchy, and it also gives the first example of a polytope with an exponential gap between equivariant psd lifts and equivariant linear programming lifts.

Keywords: semidefinite programming; extended formulations; sums of squares (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (1)

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